Elsevier

Icarus

Volume 161, Issue 2, February 2003, Pages 431-455
Icarus

Regular article
Oligarchic growth of giant planets

https://doi.org/10.1016/S0019-1035(02)00043-XGet rights and content

Abstract

Runaway growth ends when the largest protoplanets dominate the dynamics of the planetesimal disk; the subsequent self-limiting accretion mode is referred to as “oligarchic growth.” Here, we begin by expanding on the existing analytic model of the oligarchic growth regime. From this, we derive global estimates of the planet formation rate throughout a protoplanetary disk. We find that a relatively high-mass protoplanetary disk (∼10 × minimum-mass) is required to produce giant planet core-sized bodies (∼10 M) within the lifetime of the nebular gas (≲10 million years). However, an implausibly massive disk is needed to produce even an Earth mass at the orbit of Uranus by 10 Myrs. Subsequent accretion without the dissipational effect of gas is even slower and less efficient. In the limit of noninteracting planetesimals, a reasonable-mass disk is unable to produce bodies the size of the Solar System’s two outer giant planets at their current locations on any timescale; if collisional damping of planetesimal random velocities is sufficiently effective, though, it may be possible for a Uranus/Neptune to form in situ in less than the age of the Solar System. We perform numerical simulations of oligarchic growth with gas and find that protoplanet growth rates agree reasonably well with the analytic model as long as protoplanet masses are well below their estimated final masses. However, accretion stalls earlier than predicted, so that the largest final protoplanet masses are smaller than those given by the model. Thus the oligarchic growth model, in the form developed here, appears to provide an upper limit for the efficiency of giant planet formation.

Introduction

The initial growth mode in a disk of accreting planetesimals is runaway growth (e.g., Wetherill and Stewart 1989, Kokubo and Ida 1996, where the mass doubling time for the largest bodies is the shortest. However, when these runaway bodies, or protoplanets, become sufficiently massive, it is their gravitational scattering (often called viscous stirring) which dominates the random velocity evolution of the background planetesimals, rather than the interactions among the planetesimals. Since the accretion cross-section of a protoplanet is smaller among planetesimals with higher random velocities, protoplanet growth now switches to a slower, self-limiting mode, in which the mass ratio of any two protoplanets at adjacent locations in the disk approaches unity over time. Ida and Makino (1993) investigated this transition analytically and through N-body simulations, and Kokubo and Ida 1998, Kokubo and Ida 2000, Kokubo and Ida 2002 studied the subsequent accretion mode, giving it the name “oligarchic growth.” In the terrestrial region, the final accretion phase likely consisted of the merging of oligarchically accreted protoplanets; simulations show that such a process fairly readily produces bodies with masses comparable to present-day terrestrial planets (e.g., Chambers and Wetherill, 1998). However, an analogous phase in the trans-saturnian region would have been highly inefficient; even sub-Earth-mass protoplanets excite each other to high random velocities on a timescale short compared to their collision timescale, so that only negligible accretion occurs (Levison and Stewart, 2001). Thus, it appears that oligarchic growth alone must be called upon to account for almost all accretional growth in the outer Solar System.

In Section 2, we examine the condition for crossover from runaway to oligarchic growth and show that this transition is expected to set in when the largest bodies are still several orders of magnitude below an Earth mass. In Section 3, we summarize the previous work on oligarchic growth timescale estimates, obtain a protoplanet mass function, and then extend the model by considering a system in which the planetesimal surface density varies in a self-consistent way. We show that with an approximately 10-fold increase in surface density relative to the minimum-mass model, protoplanets of mass ∼10 M can form. The standard nucleated instability model of gas giant formation, wherein a massive gas envelope accumulates onto a solid core during the nebular gas lifetime (≲10 million years; e.g., Strom et al., 1990), is thought to require bodies of this mass Mizuno et al 1978, 1996. Our estimate of the required density enhancement is somewhat higher than that of Weidenschilling (1998), who finds, using a multizone statistical simulation, that 4× the minimum mass is insufficient to produce giant planet cores, but that an additional “modest increase” is sufficient to make it happen. The formation of “ice giant” planets such as Uranus and Neptune at stellocentric distances of ≳20 AU cannot be similarly accounted for during this time.

In Section 4, we consider the effect of planetesimal size on the accretion rate and accretion efficiency. In Section 5, we discuss the validity of the model. In Section 6, we obtain oligarchic growth rate estimates in the absence of gas, to ascertain how much more accretion could have taken place subsequent to the removal of the nebular gas. We consider two extremes: that of collisionless planetesimals, and that of maximally effective collisional damping of random velocities (though without fragmentation). In the former limit, Uranus- and Neptune-mass planets cannot be produced at their current locations on any timescale unless the initial protoplanetary disk is implausibly massive; in the latter limit, such planets might be formed in a reasonable-mass disk and in less than the age of the Solar System. In Section 7, we test the semianalytic predictions for the pre-gas-dispersal phase of oligarchic growth against numerical simulations (parameters in Table 1). We find good agreement as long as protoplanet masses are well below their theoretical final masses; however, growth in the simulations stalls early, so that the final masses fall short of those predicted by the model. We summarize the results and discuss implications in Section 8.

Section snippets

Transition to oligarchic growth

Ida and Makino (1993) derive the following condition for the dominance of protoplanet–planetesimal scattering over planetesimal–planetesimal scattering in determining the random velocity evolution of the planetesimal disk, 2∑MM>∑mm, where M and m are the protoplanet mass and the effective planetesimal mass, respectively, Σm is the surface mass density of the planetesimal disk, and ΣM is the effective surface density of a protoplanet in the disk. The last is given by M=M2πaΔastir where a is

Oligarchic growth rate estimates

When planetesimal random velocities are dispersion-dominated rather than shear-dominated, the mass accretion rate of an embedded protoplanet is well described by the particle-in-a-box approximation Safronov 1969, Wetherill 1980, Ida and Nakazawa 1989, dMdt≃F mhπ RM2 1+υesc2υrel2 υrel, where h the disk scale height, RM the protoplanet radius, υesc the escape velocity from the protoplanet’s surface, and υrel the characteristic relative velocity between the protoplanet and the planetesimals. F

The effect of planetesimal size

The efficiency of protoplanet accretion in this model is subject to two competing effects, both of which, for a given nebular gas density, depend on the characteristic planetesimal size. On the one hand, smaller planetesimals experience stronger damping of random velocities, forming a thinner disk and thus increasing the accretion rate. On the other hand, smaller planetesimals are also subject to faster orbital decay, which depletes the planetesimal surface density at a given location in the

Validity of the estimate

A number of simplifications underlie this estimate of protoplanet growth rates. To begin with, interactions among planetesimals are neglected altogether in our analysis. This seems reasonable since, by definition, scattering by protoplanets dominates the planetesimal velocity distribution in the oligarchic regime. Also, Kokubo and Ida (2002) showed that the timescale for spreading of the planetesimal disk due to mutual interactions is large compared to the accretion timescale (though the disk

Oligarchic growth in the absence of gas

We have established above that collisional damping of random velocities is of little importance in the oligarchic growth regime while the gas is present. However, once the gas is removed, this may no longer be true. The issue of postgas accretion is of particular interest in the case of our Solar System, since, from Section 3, the in situ formation of Uranus and Neptune during the gas lifetime appears to be ruled out. Assuming the problem of the ice giants’ gas content can be otherwise solved,

Numerical simulations

A simple semianalytic estimate for protoplanet mass as a function of time throughout a protoplanetary disk is a potentially powerful tool, since it offers the possibility of characterizing accretional evolution over time and distance scales which are as yet beyond the reach of numerical simulation. Nevertheless, to assess the validity of such an estimate, comparisons to simulations must be made. The limits of computing capacity restrict the domains of full N-body simulations to relatively

Conclusions

Runaway growth allows very short formation times, but only in the early stages of planetesimal accretion; there is a transition to the self-limiting oligarchic growth mode when the largest bodies are still orders of magnitude below an Earth mass. The timescale of oligarchic growth thus dominates over that of runaway growth, and we use the former alone to obtain a global picture of planet formation throughout a protoplanetary disk. In the terrestrial region, accretion efficiency is high, and

Acknowledgements

This work is supported by the Center for Integrative Planetary Science (E.W.T.), NASA’s Origins of Solar Systems (E.W.T., H.F.L.), Planetary Geology Geophysics, and Exobiology (H.F.L.) programs, and by Canada’s National Science and Engineering Research Council (M.J.D.). We thank the referees, Satoshi Inaba and Eiichiro Kokubo, for valuable suggestions which helped us to improve the manuscript. We also thank Glen Stewart and Andrew Youdin for helpful discussions.

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